I don't know for sure, but there may indeed be a downside. In my case, a few years ago, I applied for an individual policy and was denied (I was shocked). When I applied to another company, they asked if I'd ever been denied and why. It didn't look good on the application!
What finally worked for me was to talk with a local insurance agent who represented multiple companies and could recommend which companies to apply to, and tell me before I applied if I'd likely be denied or not, and how much it would cost if I was accepted. If you've never applied for private insurance, you may be unpleasantly surprised at what they require you do just to apply. You don't want to put out that much effort and lose every shred of your privacy if you have no chance of being accepted anyway.
After I got accepted and had coverage for a couple of years, my insurer (Mutual of Omaha) sent me a letter telling me they were getting out of the health insurance business in my state!
I went back to the agent, and she informed me that the state I live in offers insurance for those denied coverage by private insurers. I had no idea there was that option. Spendy, but it filled the gap for me until I could retire and get coverage through an earlier employer.
My agent didn't charge me anything, other than she probably got part of my premium for those couple years I was insured.
Anyway, an insurance agent helped me here in Alaska.